Question: Simplify; express your answer in exponential form. Assume $y\neq 0, z\neq 0$. $\dfrac{{(yz^{3})^{5}}}{{(y^{-5}z^{3})^{-2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(yz^{3})^{5} = (y)^{5}(z^{3})^{5}}$ On the left, we have ${y}$ to the exponent ${5}$ . Now ${1 \times 5 = 5}$ , so ${(y)^{5} = y^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(yz^{3})^{5}}}{{(y^{-5}z^{3})^{-2}}} = \dfrac{{y^{5}z^{15}}}{{y^{10}z^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{5}z^{15}}}{{y^{10}z^{-6}}} = \dfrac{{y^{5}}}{{y^{10}}} \cdot \dfrac{{z^{15}}}{{z^{-6}}} = y^{{5} - {10}} \cdot z^{{15} - {(-6)}} = y^{-5}z^{21}$